Once Socrates' philosophical method began to inspire potentially disruptive questions and changes in the young people of Athens, the people in power become alarmed. They put Socrates on trial for corrupting minds of the youth and destabilizing the social order in Athens.
As is so often the case, once a person such as Socrates starts making an actual difference, that person is shut down as quickly as possible, often through death. The irony of the statement that the unexamined life is not worth living is that it is not for examining his life that Socrates got into trouble: it was through being active in the community by inciting too many other people to do the same.
I agree that an unexamined life is not worth living, but I would emphasize what is implicit in Socrates' statement: that an examined life is primarily worthwhile if it leads to change. Other traditions have understood this: the early Protestants were often fixated on examination of conscience in order to make themselves more...
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